We study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These metrics originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.

翻译：我们研究了有界增长可积函数空间上的Hilbert射影度量变体。这些度量源于非负函数锥的松弛形式——具体而言，它们包含所有与特定检验函数相乘后积分值非负的函数。我们证明：即使核函数在零点处无下界，只要其衰减速率受控，核积分算子在这些度量的适当设定下仍具有压缩性。作为熵正则化最优传输的应用，我们证明当边际分布尾部衰减速度显著快于成本函数增长时，Sinkhorn算法具有指数收敛性。